#8384927

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19,22,24)(18,23,26,20)(21,27,29,31)(25,30,32,28)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,8)(5,7)(9,12)(14,16)(21,29)(25,32)(27,31)(28,30)(34,38)(39,41)(40,44)(47,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,32)(29,30)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8292123);